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Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduleval | ⊢ ◡ ≤ = (le‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . . 5 ⊢ (le‘𝑂) ∈ V | |
2 | 1 | cnvex 7612 | . . . 4 ⊢ ◡(le‘𝑂) ∈ V |
3 | pleid 16659 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
4 | 3 | setsid 16530 | . . . 4 ⊢ ((𝑂 ∈ V ∧ ◡(le‘𝑂) ∈ V) → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
5 | 2, 4 | mpan2 690 | . . 3 ⊢ (𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
6 | 3 | str0 16527 | . . . 4 ⊢ ∅ = (le‘∅) |
7 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (le‘𝑂) = ∅) | |
8 | 7 | cnveqd 5710 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ◡∅) |
9 | cnv0 5966 | . . . . 5 ⊢ ◡∅ = ∅ | |
10 | 8, 9 | eqtrdi 2849 | . . . 4 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ∅) |
11 | reldmsets 16503 | . . . . . 6 ⊢ Rel dom sSet | |
12 | 11 | ovprc1 7174 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
13 | 12 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) = (le‘∅)) |
14 | 6, 10, 13 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
15 | 5, 14 | pm2.61i 185 | . 2 ⊢ ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
16 | oduval.l | . . 3 ⊢ ≤ = (le‘𝑂) | |
17 | 16 | cnveqi 5709 | . 2 ⊢ ◡ ≤ = ◡(le‘𝑂) |
18 | oduval.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
19 | eqid 2798 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
20 | 18, 19 | oduval 17732 | . . 3 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
21 | 20 | fveq2i 6648 | . 2 ⊢ (le‘𝐷) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
22 | 15, 17, 21 | 3eqtr4i 2831 | 1 ⊢ ◡ ≤ = (le‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ◡ccnv 5518 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 lecple 16564 ODualcodu 17730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16478 df-slot 16479 df-sets 16482 df-ple 16577 df-odu 17731 |
This theorem is referenced by: oduleg 17734 odupos 17737 oduposb 17738 oduglb 17741 odulub 17743 posglbd 17752 oduprs 30669 odutos 30676 mcgcnv 30705 ordtcnvNEW 31273 ordtrest2NEW 31276 |
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